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A simple proof of a Kramers’type law for self-stabilizing
diffusions
Julian Tugaut
To cite this version:
A simple proof of a Kramers’type law for
self-stabilizing diffusions
Julian Tugaut
**
Universit´e Jean Monnet, Institut Camille Jordan, 23, rue du
docteur Paul Michelon, CS 82301, 42023 Saint-´
Etienne Cedex 2,
France.
March 5, 2015
Abstract
We provide a new proof of a Kramers’ type law for self-stabilizing diffusion. These diffusions correspond to the hydrodynamical limit of a mean-field system of particles and may be seen as the probabilistic inter-pretation of the granular media equation. We use the same hypotheses as the ones used in the work “Large deviations and a Kramers’ type law for self-stabilizing diffusions” by Herrmann, Imkeller and Peithmann in which the authors obtain a first proof of the statement.
Key words and phrases: Self-stabilizing diffusion ; Exit time ; Large deviations ; Coupling method
2000 AMS subject classifications: Primary: 60F10 ; Secondary: 60J60, 60H10
1
Introduction
In their remarkable work “Large deviations and a Kramers’ type law for self-stabilizing diffusions”, Herrmann, Imkeller and Peithmann establish large devi-ation results and solve the exit problem of the so-called self-stabilizing diffusion. This consists of the following model.
Xtǫ= X0+ Z t 0 V (Xs)ds −ǫ Z t 0 Z Rd Φ (Xsǫ− x) duǫs(x)ds + √ ǫWt. (1) In this equation, V and Φ denote vector fields on Rd; (Wt)t≥0is a d-dimensional Wiener process ; duǫ
Equation (1) corresponds to the hydrodynamical limit of a mean-field system of particles. Ztǫ,i,N = X0+ √ ǫWti− Z t 0 V Zsǫ,i,N ds − 1 N N X j=1 Z t 0 Φ Zsǫ,i,N− Zsǫ,j,N ds
for all 1 ≤ i ≤ N. Here, the Wi are independent Brownian motions and W1= W . See [Szn91].
In [HIP08], the authors consider an open domain D satisfying some hypothe-ses and they study the limit as ǫ goes to 0 of ǫ log {E [τD(ǫ)]} where τD(ǫ) is defined as the first exit time of Xǫ from the domain D:
τD(ǫ) := inf {t ≥ 0 | Xǫ t ∈ D} ./ More precisely, they obtain the limit
lim ǫ→0 Pne1ǫ(Q∞−ξ) < τD(ǫ) < e 1 ǫ(Q∞+ξ) o = 1 , for any ξ > 0. Here, Q∞ denotes the exit cost of the domain D:
Q∞:= inf z∈∂DT >0inf ϕ∈Hinf1 z 1 2 Z T 0 || ˙ϕt− V (ϕt) + Φ(ϕt− xstable)|| 2 dt . (2) The set H1
z denotes the space of absolutely continuous functions f such that f (0) = xstable and f (T ) = z. And, xstable is the unique point in which the vector field V is equal to 0.
In a gradient case, we simply have Q∞ := 2 infz∈∂D(W (z) − W (xstable)), where the potential W is defined by ∇W (x) = V (x) − Φ(x − xstable).
To obtain their result, they reconstruct Freidlin-Wentzell theory to the self-stabilizing diffusion. More recently, we published a work ([Tug12]) which proves the same result (albeit only in the gradient case) by using a different method. We solve the exit problem of the first particle Zǫ,1,N and we use a coupling method between Zǫ,1,N and Xǫ. The proof is more natural and intuitive but is more technical.
The aim of this paper is to provide a much simpler method to obtain the result. For a complete review of Freidlin-Wentzell theory, see [DZ98, FW98].
First, we give the assumptions of the paper, that are the same as the ones in [HIP08] (see pages 1383 and 1406). Then, we remind the reader the main result of the paper that concerns the exit time. We have chosen not to deal with the exit location because we do not have improvement of this question. In a third section, we provide the proof of the theorem.
2
Assumptions and notations
Assumption (A): We say that the coefficients V and F satisfy the set of assumptions (A) if
(A-1) The coefficients V and Φ are locally Lipschitz, that is, for each R > 0 there exists KR > 0 such that ||V (x) − V (y)|| + ||Φ(x) − Φ(y)|| ≤ KR||x − y||, for x, y ∈ BR(0) :=z ∈ Rd : ||z|| < R .
(A-2) The interaction function Φ is rotationally invariant, that is, there exists a function φ from [0; +∞[ to [0; +∞[ such that Φ(x) = ||x||x φ(||x||), x 6= 0. (A-3) The function φ is convex and φ(0) = 0.
(A-4) The function Φ grows at most polynomially: there exists K > 0 and r ∈ N such that ||Φ(x) − Φ(y)|| ≤ ||x − y|| (K + ||x||r+ ||y||r), x, y ∈ Rd. (A-5) The function V is continuously differentiable.
(A-6) The vector field V is convex: let DV (x) denote the Jacobian of V . We assume that there exists KV > 0 such that hh ; DV (x)hi ≤ −KV, for h ∈ Rd such that ||h|| = 1 and x ∈ Rd.
(A-7) We assume that the unique point in which the vector field is equal to 0 is xstable.
Under these seven assumptions, here exists a positive integer n0 such that supx∈K|Φ ∗ µ(x) − Φ(x − xstable)| ≤ K (M1, · · · , Mn0), K being a continuous
function such that K(0, · · · , 0) = 0 and Mp:=R
Rd||y − xstable||
p µ(dy). We now present the definition of what we denote as “stable by”.
Definition 2.1. Let k be any positive integer. Let G be a subset of Rk and let U be a vector field from Rk to Rk which satisfies the set of assumptions (A). For all x ∈ Rk, we consider the dynamical system ψt(x) = x +Rt
0U (ψs(x)) ds. We say that the domain G is stable by U if the orbit {ψt(x) ; t ∈ R+} is included in G for all x ∈ G.
Hypothesis 2.2. We consider the dynamical system ϕt = X0+Rt
0V (ϕs) ds where X0 is introduced in (1). The orbit {ϕt; t > 0} is included in D.
Hypothesis 2.3. The open domain D is stable by V − Φ(. − xstable). Definition 2.4. B∞
κ denotes the set of all the probability measures µ on Rd satisfyingR
Rd||x − xstable||
2n
µ(dx) ≤ κ2n.
We now give the main result of the current work.
Theorem: We consider vector fields V and Φ which satisfy the set of assump-tions (A). Under Assumpassump-tions 2.2–2.3, for all ξ > 0, we have the limit:
lim ǫ→0P n e1ǫ(Q∞−ξ) < τD (ǫ) < e1ǫ(Q∞+ξ)o = 1 .
Let us notice that Herrmann, Imkeller and Peithmann assume a stronger hypothesis than Hypothesis 2.3. Indeed, in our work, the domain D is not assumed to be stable by V .
3
Proof of Theorem
3.1
Control of the moments
We now establish an important result about the moments of Xǫ. Indeed, since these moments intervene in the drift, the asymptotic behaviour (determinis-tic) of the law uǫ
t is related to the asymptotic behaviour (probabilistic) of the trajectories.
Property 3.1. 1. The 2nth moment is uniformly bounded: sup t∈R+ En||Xtǫ||2no≤ max ||X0||2n ; 2n − 1 2KV n ǫn . (3) 2. For all κ > 0 and ǫ > 0, we introduce the deterministic time
Tκ(ǫ) := minnt ≥ 0 E n ||Xtǫ|| 2no ≤ κ2no. For ǫ <κ2KV
2n−1, we have the inequality: Tκ(ǫ) ≤ 1
nKVκ2n||X0||
2n . 3. Moreover, for all t ≥ Tκ(ǫ), En||Xǫ
t|| 2no ≤ κ2n. Proof. We put ξǫ(t) := En||Xǫ t|| 2no
. We apply the Itˆo formula, we integrate, we take the expectation then we take the derivative. We obtain:
ξ′ ǫ(t) =2nE n ||Xǫ t|| 2n−2 hXǫ t; V (Xtǫ)i o − 2nEn||Xǫ t|| 2n−2 hXǫ t; Φ ∗ uǫt(Xtǫ)i o + n(2n − 1)ǫEn||Xǫ t|| 2n−2o =: 2n (aǫ(t) + bǫ(t)) + cǫ(t) . By definition, the second term bǫ(t) can be written as
bǫ(t) = Eh||Xtǫ|| 2n−2
hXtǫ; Φ (Xtǫ− Ytǫ)i i
where Yǫ is a solution of (1) independent from Xǫ. We can exchange Xǫ and Yǫ. Thereby, by using the assumptions, we get:
bǫ(t) = E φ (||X ǫ t − Ytǫ||) ||Xǫ t− Ytǫ|| D ||Xt||ǫ 2n−2 Xǫ t; Xtǫ− Ytǫ E = 1 2E φ (||Xǫ t − Ytǫ||) ||Xǫ t− Ytǫ|| D Xǫ t||Xtǫ|| 2n−2 − Yǫ t ||Ytǫ|| 2n−2 ; Xǫ t − Ytǫ E . This last term is nonnegative. Indeed, the Cauchy-Schwarz inequality implies
D
x ||x||2n−2− y ||y||2n−2; x − yE≥||x||2n−1− ||y||2n−1(||x|| − ||y||) ≥ 0 for all x, y ∈ Rd. Therefore, we obtain bǫ(t) ≥ 0.
Hence, by using Jensen inequality, we deduce cǫ(t) ≤ n(2n − 1)ǫξǫ(t)1− 1 2n. By
combining results on aǫ(t), bǫ(t) and cǫ(t), we obtain ξǫ(t) ≤ −2nKV′ ξǫ(t)1−n1 ξǫ(t)n1 −(2n − 1)ǫ 2KV . (4)
The statements of the lemma are obvious consequences of Inequality (4). This means that the self-stabilizing process tends to be trapped in a ball with center 0 = xstable.
3.2
Probability of exiting before T
κ(ǫ)
In this paragraph, we give the following result.
Property 3.2. We have the limit: limǫ→0P(τD(ǫ) < Tκ(ǫ)) = 0 for any κ > 0. We skip the proof but the ideas are the following. For all δ > 0, we introduce
τδ(ǫ) := inf {t > 0 : ||Xǫ
t− ϕt|| > δ} , where we remind the reader that ϕt= X0+Rt
0V (ϕs) ds. Thus, for any T > 0, the following limit is an easy and classical result: limǫ→0P(τδ(ǫ) < T ) = 0. However, here, we consider the interval [0; Tκ(ǫ)] which depends on ǫ. But, we have uniformly bounded Tκ(ǫ). Indeed, we have
P(τδ(ǫ) < Tκ(ǫ)) ≤ P τδ(ǫ) < 1 nKVκ2n ||X0|| 2n , which goes to 0 as the noise elapses.
Due to hypothesis 2.2, we have {ϕt : t > 0} ⊂ D. Consequently, for any κ > 0, we obtain the limit limǫ→0P(τD(ǫ) < Tκ(ǫ)) = 0.
3.3
Coupling result
Let K be a compact domain which contains the open set D.
We have proven that the diffusion does not exit the domain D before the time Tκ(ǫ). Now, we study the exit of the diffusion from the domain after the time Tκ(ǫ). To do so, we use the following fact: supt≥Tκ(ǫ)E
n ||Xǫ
t|| 2no
≤ κ2n. Since this inequality holds for any κ > 0, we deduce that the drift V − Φ ∗ uǫ t is close to the vector field V − Φ ∗ δ0 = V − Φ. Consequently, we consider the following diffusion defined for t ≥ Tκ(ǫ):
Ytǫ= XTκ(ǫ)+ √ ǫ Wt− WTκ(ǫ) + Z t Tκ(ǫ) V (Ysǫ)ds − Z t Tκ(ǫ) Φ (Ysǫ) ds , (5) if XTκ(ǫ) ∈ K and Y ǫ
t := Xtǫotherwise. We introduce the two following exit time: τK(ǫ) := inf {t > Tκ(ǫ) : Xǫ
Theorem 3.3. There exists κ0 such that for all κ < κ0, there exists ǫ0(κ) > 0 such that PnsupTκ(ǫ)≤t≤TK,κ(ǫ)||X
ǫ
t− Ytǫ|| ≥ r(κ) o
≤ r(κ) for all ǫ < ǫ0(κ). Here, r is a positive and increasing function such that r(0) = 0.
Proof. Step 1. We introduce the vector fields H∞(x) := V (x) − Φ(x) and Ht(x) := V (x) − Φ ∗ uǫ
t(x). The assumptions on V and Φ imply DHt(x) ≤ −KV < 0. From now on, we put ξǫ(t) := ||Xǫ
t− Ytǫ||. If XTǫκ, Y ǫ Tκ ∈ K then, for all Tκ≤ t ≤ Tκ(ǫ), we have: d dt(ξǫ(t)) 2 = − 2 hHt(Xt) − H∞ǫ (Yǫ t) ; Xtǫ− Ytǫi = − 2 hHt(Xt) − Htǫ (Ytǫ) ; Xtǫ− Ytǫi − 2 hΦ ∗ uǫ t(Ytǫ) − Φ (Ytǫ) ; Xtǫ− Ytǫi ≤ − 2KV(ξǫ(t))2+ 2ξǫ(t)fK(κ) , (6) where we set fK(κ) := supµ1∈B∞
κ supx∈K||Φ ∗ µ1(x) − Φ(x)|| =: KVr(κ) 3 2.
In-equality (6) directly implies supTκ≤t≤Tκ(ǫ)||X
ǫ t− Ytǫ|| 2 ≤ r(κ)3 which yields EnsupT κ(ǫ)≤t≤Tκ(ǫ)||X ǫ t− Ytǫ|| 2o
≤ r(κ)3. The claim thus follows from the Markov inequality.
3.4
Proof
Step 1. Let κ > 0. We can easily prove (by proceeding like in [Tug12]) that there exist two families of domains (Di,κ)κ>0 and (De,κ)κ>0 such that
• Di,κ⊂ D ⊂ De,κ.
• Di,κand De,κare stable by V − Φ. The terminology “stable by” has been introduced in Definition 2.1.
• sup z∈∂Di,κ
d (z ; Dc) + sup z∈∂De,κ
d (z ; D) tends to 0 when κ goes to 0. • inf
z∈∂Di,κ
d (z ; Dc) = inf z∈∂De,κ
d (z ; D) = r(κ).
Step 2. By τi,κ(ǫ) (resp. τe,κ(ǫ)), we denote the first exit time of Yǫfrom Di,κ (resp. De,κ).
Step 3.1. By classical results in Freidlin-Wentzell theory, there exists κ1 > 0 such that for all 0 < κ < κ1, we have: limǫ→0Pτe,κ(ǫ) < exp1
ǫ Q∞+ ξ = 0. Therefore, the first term aκ(ǫ) tends to 0 as ǫ goes to 0.
Step 3.2. Let us look at the second term bκ(ǫ). For κ sufficiently small, we have De,κ⊂ K. Consequently, we have:
Pnτ (ǫ) ≥ eQ∞ +ξǫ ; τe,κ(ǫ) ≤ e Q∞ +ξ ǫ o ≤ Pn X ǫ τe,κ(ǫ)− Y ǫ τe,κ(ǫ) ≥ r(κ) o ≤ P ( sup Tκ(ǫ)≤t≤TK,κ(ǫ) ||Xtǫ− Ytǫ|| ≥ r(κ) ) . According to Theorem 3.3, there exists ǫ0 > 0 such that the previous term is less than r(κ) for all ǫ < ǫ0.
Step 3.3. Let ξ > 0. By taking κ arbitrarily small, we obtain the upper bound limǫ→0Pnτ (ǫ) ≥ exphQ∞+ξ
ǫ io
= 0.
Step 3. Analogous arguments show that limǫ→0PnTκ(ǫ) ≤ τ(ǫ) ≤ eQ∞ −ξǫ
o = 0. However, we have limǫ→0P{τ(ǫ) ≤ Tκ(ǫ)} = 0. This ends the proof.
Remark 3.4. In the theorem, we give the exit time of the McKean-Vlasov diffusion but we could use the same technics to provide the exit time of the first particle in the mean-field system of particles. The only difference is that we would need to use the first time that the empirical measure exits from the ball B∞κ ; which is close to the arguments in [Tug12].
Acknowledgements
I would like to thank Samuel Herrmann for having introduced me to the problem. Velika hvala Marini za sve. ´Egalement, un tr`es grand merci `a Manue, `a Sandra et `a Virginie pour tout.
References
[DZ98] A. Dembo and O. Zeitouni: Large deviations techniques and applications, volume 38 of Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2010. Corrected reprint of the second (1998) edition.
[FW98] M. I. Freidlin and A. D. Wentzell: Random perturbations of dynamical systems, volume 260 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, second edition, 1998. Translated from the 1979 Russian original by Joseph Sz¨ucs.
[HIP08] Samuel Herrmann, Peter Imkeller, and Dierk Peithmann. Large deviations and a Kramers’ type law for self-stabilizing diffusions. Ann. Appl. Probab., 18(4):1379–1423, 2008.
[Szn91] Alain-Sol Sznitman. Topics in propagation of chaos. In ´Ecole d’ ´Et´e de Probabilit´es de Saint-Flour XIX—1989, volume 1464 of Lecture Notes in Math., pages 165–251. Springer, Berlin, 1991.
